Talk:Complex number/Draft

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See comments above

I made some comments in the section Talk:Complex number/Draft#What the symbol i means in this article above, marked with bullet points, which have not yet been addressed. (I just wanted to mention them after the page break so they won't be forgotten when editing the next version.) --Catherine Woodgold 20:28, 6 May 2007 (CDT)

I edited in all of the changes I had suggested except for the stuff about quantum physics. I don't have a textbook on the subject handy. --Catherine Woodgold 19:17, 7 May 2007 (CDT)

I'm not entirely happy with my text in the QM section, either. Trying to interpret superposed states in terms of probabilities is dicey at best, anyway. I'll have to think about this and see if I can come up with something better. In any case, I'm intrigued by what Robert Tito had to say about other uses of complex numbers, particularly in Hamiltonian systems (conjugate coordinates with a factor of i?) Anyway, I was just trying to come up with something that would be recognizable to a wide range of readers (albeit not mathematically naïve ones). If nothing else, the Schrödinger Equation has a certain iconic value. I'm certainly open to other suggestions. Greg Woodhouse 19:38, 7 May 2007 (CDT)

Wait! I think the quantum physics stuff is good! It just needs some editing, as I suggested, e.g. defining the symbols used etc. --Catherine Woodgold 19:42, 7 May 2007 (CDT)

It seems to me that saying things like "$\hbar$ is Planck's constant divided by $2\pi$" wouldn't really add anything to the article, and I guess that's what bothers me: if you (generic) know what Schrödinger's equation is, this probably doesn't need to be said, and if you don't, the section really doesn't add anything. Greg Woodhouse 22:40, 7 May 2007 (CDT)

Part of chemistry and physics workgroups?

(I hope no one minds if I move this discussion "below the bar". Greg Woodhouse 03:34, 7 May 2007 (CDT))

I am just curious why this article's checklist includes it in the chemistry and physics workgroups. It seems that even though this article has applications in those field, including it in every workgroup it applies could get out of hand. - Jared Grubb 12:32, 6 May 2007 (CDT)

that answer is simple: the need for something like a complex number arose from these sciences not from math. Math formalized it, thats all. Robert Tito |  Talk 

I disagree. We have an example showing that complerx numbers are important in the sciences, too, but complex numbers were intoduced in a fundamental way in mathematics (i.e., not just as a notational convenience) long before quantum mechanics had even been thought of. Greg Woodhouse 16:59, 6 May 2007 (CDT)

Then again, since those workgroups are there, maybe you can sign off on it, too. :-) Greg Woodhouse 17:02, 6 May 2007 (CDT)

Physics and chemistry used the notion of compplex numbers as from the 18th century - when they needed them to describe things. Euler, Gauss, Fourier are not mathematicians but physicists/chemists that needed a solution for their math problems. the complext number by far didn't start with quantum mechanics. I might mention Hamiltonian mechanics as an example, or canonicals. Robert Tito |  Talk  17:35, 6 May 2007 (CDT)

It still seems a little odd to me. But, I suppose Ohm's law would be more at home in the electrical engineering workgroup than the mathematics, even though it is a mathematic equation... - Jared Grubb 23:14, 6 May 2007 (CDT)

The historical development of the concept of complex numbers seems like an interesting topic for an article (albeit a challenging one!), but so far as this article is concerned, I don't think it's really that important. No, that doesn't sound right: I don't mean it's not important, only that I don't think it needs to be addressed in the context of this article. Greg Woodhouse 03:34, 7 May 2007 (CDT)

Does anyone know how to create an archive? Is there an automated, or at least "official" way to do it? I just got the following warning:

WARNING: This page is 87 kilobytes long; some browsers may have problems editing pages approaching or longer than 32kb. Please consider breaking the page into smaller sections.

Greg Woodhouse 03:34, 7 May 2007 (CDT)

Re archiving: you might want to discuss with Chris Day or see Talk:Biology/Draft, but since that template is named "Experimental" I suppose procedures haven't been finalized. (Discussion about it arising from pages like this one may drive the finalization of such procedures.) Maybe it's being discussed on the forum somewhere, or if not someone could start. --Catherine Woodgold 07:45, 7 May 2007 (CDT)

Re being part of chemistry and physics workgroups: Chemistry and physics also need to use 1 + 1 = 2. They also need to use words with syllables to communicate technical concepts; that doesn't mean the linguistics Syllable page has to be in the chemistry and physics workgroups. Those sciences use math -- that doesn't mean math is part of the science. I think perhaps people in the chemistry and physics workgroups should decide whether the article is included or not. It's OK with me either way -- it's not that unreasonable. More justifiable than including a page that presents a proof of 1 + 1 = 2 in those sciences. --Catherine Woodgold 18:23, 7 May 2007 (CDT)

This article was approved by a math editor and currently is listed in the Math Workgroup Approved articles, but not in Chemistry or Physics. Since the article is cross-listed in three workgroups, will there need to be three approval processes? Or will we need editors from all three areas to agree before any one draft gets approved? Or will we declare one "father" workgroup, and the others just raise objections or not... I know this approval process is still in its infancy, but these are questions we really should address at some point. - Jared Grubb 02:41, 8 May 2007 (CDT)

Error in multiplicative property

I've just put the following message on User talk:Nancy Sculerati.

Dear Nancy. Etienne Parizot found and fixed an error in Complex number/Draft which is also present in the approved version, Complex number. The formula

$e^{z_{1}z_{2}}=e^{z_{1}}e^{z_{2}}\$

halfway the section "The complex exponential" should read

$e^{z_{1}+z_{2}}=e^{z_{1}}e^{z_{2}}\$

(with a plus sign added on the left-hand side).

This is a big error so I want it to be fixed as soon as possible. I can't imagine any editor would argue with this change. However, I'm not sure what our options are.

• Some places hint at the possibility to have the constabulary do limited changes to articles without going through the whole approval process (for instance, the section #Copyediting matters above). I couldn't find anything about rules or procedure though. If such a possibility exists, that would be my preference. For the record, the article complex number was nominated by Greg Martin and the nomination was supported by me.
• If this is not possible, I'd like the approval to be revoked. As far as I can see, there is no rule or precedent for this, only an empty section at CZ:Approval Process.
• If neither of the above is possible, or if it would take too long, we can always go for the option of nominating the fixed version for approval. To be honest, it's not that important in the big scheme of things, but it is embarrassing and I feel responsible for it.

Any guidance from you (or anybody who happens to read this) would be much appreciated.

-- Jitse Niesen 08:37, 10 May 2007 (CDT)

Hi All, I have commented out the Approval tag per nominating editor Jitse Niesen who has revoked his approval. During this time, our Approval editor, Nancy Sculerati can make the appropriate changes and she can replace the Approval tag. If more chances are made, then I would suggest giving yourself an additional 24 hours before re-approval to give others a chance to review the changes. --Matt Innis (Talk) 08:46, 10 May 2007 (CDT)

I think this was handled very well. What I saw was: Etienne correctly decided that this problem should be corrected very quickly; Nancy sent me (presumably other editors) an email to alert me to it; by the time I came to CZ, it had already been decided that the change was appropriate and needed, and was made to the approved version as well as to the draft.

Although this might be "outside the rules", I think that here judgment wins the day, concerning what we might call "clear factual mistakes or obvious typos". I certainly take responsibility for nominating the article for approval without seeing this mistake. And sharp eyes Etienne! - Greg Martin 14:29, 10 May 2007 (CDT)

There will always be such mistakes, and a good approval process can take care of them.It is understandable that Jitse, who was shocked by the sudden recognition of such a mistake, wanted it fixed IMMEDIATELY. It was understandable that Matt acted to accomodate him, my only point is -in the future we now know that the approvals editor could have done the copyedit at Jitse's say so. Approval cannot be "revoked" in this manner. Think about it. If it could be, that sets a terrible precedent, you can imagine how in a different circumstance such a precedent could be misused. I cannot add details to the approval process policy without the Editorial Council (of which I am a member) being up and running, with a voting process in place. Right now we are figuring out the process. I said several times that I would take responsibilty for copyedits at this stage with any of the nominating editors.Maybe we should add that, in an emergency the constable can put up a note saying that there is a copyediting problem that is being corrected- in progress. Nancy Sculerati 12:10, 11 May 2007 (CDT)

Style issues

Quoting some examples of style that I consider a bit too informal. In particular, there are many phrases/clauses that make the article verbose. In the first paragraph, there are: Of course, As it happens, At first glance, perhas more importantly.

Is this as per the policy of Citizendium? Should the number of such phrases/clauses be reduced? Vipul Naik 02:19, 8 June 2007 (CDT)

Hi Vipul, and welcome to CZ!, the answer to your first question is "yes, there are style differences here" - see this section of the article mechanics article concerning style. Your input is welcome. Matt Innis (Talk) 07:56, 8 June 2007 (CDT)

Personally, I prefer more casual or informal style. Of course, this doesn't mean the articles need be any less precise or rigorous, only more readable, and maybe a little less intimidating. Greg Woodhouse 11:39, 8 June 2007 (CDT)

Remaining errors in this approved article?

Hi, I came across this Error claims on WP Signpost. I think there are valid points there, especially regarding the interpretation of 1/z and the comment on the potential function (clearly it can't represent some force since it is a scalar). These should be looked at closer. Are there plans to have this article revised in the near future? Thanks. Hendra I. Nurdin 13:55, 20 October 2007 (CDT)

Hi, did you look at the draft? ;-) Except for the potential thing, deleted from the draft some time ago, I don't really think this is as problematic as suggested. But if you feel like, we could find a better wording for some text. Then re-approving looks like a good idea. Aleksander Stos 14:41, 20 October 2007 (CDT)

PS. You may also have a look at my "advanced" draft. At present I gave up the idea behind that work -- but some portions of the article might be useful here. I don't know. Aleksander Stos 16:02, 20 October 2007 (CDT)

I'm merely passing on some criticisms I happened to stumble onto (I don't know how many people have read it before). Of course, if there is anything valid in them then they should be considered. As for "complex division amounting to conjugation with scaling", well it does sound a bit misleading to me (I don't know about other people, which is why I brought it up here :-)). Consider $z_{1}/z_{2}=z_{1}/|z_{2}|^{2}{\bar {z}}_{2}$ then it does not have anything to do with the conjugate of $z_{1}z_{2}$ which is ${\bar {z}}_{1}{\bar {z}}_{2}$ -- so what does "conjugation" in this part of the article refer to? Compare this for example, with the discussion of $z_{1}z_{2}$ in the article. As for your "advanced" draft (such as the section on roots of complex numbers), perhaps parts of it can go as subpages of the article? Hendra I. Nurdin 19:35, 20 October 2007 (CDT)

P.S. Does the removal of the assertion that the potential function represents some force from the article not warrant a re-approval process? This gives rise again to the issue that some relatively "minor" changes like this to an approved article should be possible to do with ease. Hendra I. Nurdin 19:44, 20 October 2007 (CDT)

Hendra, all it would take is a math editor who has not worked on it to nominate it for re-approval, or three editors who have worked on it to re-approve. If we can get that together, I will be glad to make the draft the approved version. --Matt Innis (Talk) 20:40, 20 October 2007 (CDT)

Well, let's see what others think about this first, as it could be that it's just me being pedantic and perhaps in view of others these changes may not be necessary :-) Anyways, any further changes need to be considered carefully, so that if it does have to go through re-approval, no further minor changes would need to be made afterwards. Hendra I. Nurdin 21:08, 20 October 2007 (CDT)

Okay, sounds like a plan. I was going to leave a message on Jitse's page but you beat me to it! I'll wait and see what develops. If you have any questions, just stop by my talk page. --Matt Innis (Talk) 21:53, 20 October 2007 (CDT)

I hadn't seen the criticisms before. I think they are valid points and that we should revise the article accordingly. Hendra, please change the draft as you see fit (I'm rather busy now so I can't be of much help at the moment, sorry). The approval process is not that much effort, so you shouldn't worry about that. -- Jitse Niesen 22:07, 20 October 2007 (CDT)

Done. I have also added a remark that division by c+di is only defined if c and d are not simultaneously zero in the part of the article that discusses operation on complex numbers. Therefore I invite all authors and editors who had been previously involved in the approved article to check my edits and make any modifications and corrections as deemed necessary. However, I think that the work is not all done yet. There is a bit more to be done on the section about complex numbers in physics. The sentence

"Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ..."

is quite vague and is likely to cause misunderstanding. I guess I know a fair bit about the mathematical formalisms of quantum mechanics, but I'd rather not delete things nor make substantial changes without first soliciting the opinions of those who have worked on this part, and other authors who know the subject quite well, and get their input on what is meant exactly by this sentence and whether it needs to be further elaborated upon for clarity, or changed to avoid misinterpretations. Hendra I. Nurdin 00:25, 21 October 2007 (CDT)

QM again

The article states:

Now, there is some subtlety in the interpretation of ψ because a system can be affected by observation, and the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation, but when we do measure, say, the position of a particle, the probability of finding it in a small region R is just ...

In my view this sentence should be deleted because it has absolutely nothing to do with complex numbers. It gives me the unpleasant WP experience of something that is added by somebody somewhere with some time on his hands, which is why many WP articles are headache-causing kinds of patchwork. At most one could do in this article is a link to quantum mechanics, where the Born postulate for probability of observation can be put in proper setting. --Paul Wormer 07:42, 21 October 2007 (CDT)

PS Most interactions in QM are invariant under time reversal. It can be shown that ψ can be chosen to be real in that case. And indeed, 95% of quantum chemistry deals with real functions. --Paul Wormer 07:45, 21 October 2007 (CDT)

Yes, deletion of the whole section would be one solution. As a replacement application we could instead insert the Laplace and Fourier transforms which use complex numbers in an essential way, or perhaps something on phasors. Let's see what the editors think would be best. Hendra I. Nurdin 08:03, 21 October 2007 (CDT)

There are also certain irreducible representations of some (physically important) groups that inherently are complex (Wigner, Am. J. Math. vol 62, p. 57 1941). These could be mentioned as examples of complex numbers in physics. --Paul Wormer 09:45, 21 October 2007 (CDT)

Paul, would you be interested in putting this in the article to replace what is currently there? Btw, which group does this paper talk about? Perhaps we can work on this section together, I could insert some additional engineering applications. Let me know what you think. Thanks. Hendra I. Nurdin 07:30, 23 October 2007 (CDT)

Dear Hendra, don't you think it would be a good idea to leave it to the approving editors to correct the article? We can signal what we don't like. For instance, the following sentence in the article

the functions ψ we "see" must be eigenstates of the operator defined by the Schrödinger equation

is bordering on being wrong; a wave function can be a superposition of eigenstates, see particle in a box for a graphic example. Maybe "see" refers to a collapse of the wave function, but that would be a collapse to an eigenstate of the position operator. Further, the Schrödinger equation mentioned (time-dependent) is not an eigenvalue equation, so the term "operator defined by" is pretty inconclusive.

I am of the opinion that it is better to spend our energy on new articles, given the present vast emptiness of CZ. In Legendre polynomial I linked to orthogonal polynomials. I saw that you wrote Gram-Schmidt, so for you it would be a piece of cake to write a nice article about general orthogonal polynomials, with links to Laguerre, Hermite, Legendre, Jacobi, etc. Best wishes, --Paul Wormer 06:59, 25 October 2007 (CDT)

PS. Upon rereading the Wigner article that I mentioned earlier, I noticed that Wigner does not mention any specific groups, only characteristics of groups. But, complex numbers are essential for irreducible representations of cyclic groups and for the even-dimensional irreps of SU(2). Schur's second lemma requires the solution of a polynomial equation and hence an algebraically closed field. --Paul Wormer 06:59, 25 October 2007 (CDT)

I removed the whole QM section; there are simply too many problems with it. It would be nice if somebody could write a section on applications of complex numbers outside maths. You don't need permission of the approving editors to do so (that's why it's called a draft), but I hereby do give you permission in case you feel happier with it.

I'm not so sure what the best application would be to put in that section. Phasors is relatively easy to explain, but I think it's mainly an organizational tool and it's not essential to use complex numbers - one can just use sine and cosine. However, Laplace transforms may be too difficult, given that we tried hard to make the page understandable with a minimum of prior knowledge. Or perhaps QM is a good example after all when written up properly; we can just show the Schrodinger equation and say that it has an i in there.

By the way, I moved from Australia to England and that's why I haven't been around much lately. Still settling in, and all my books are still en route, but I should be able to spend some more time here soon. -- Jitse Niesen 07:51, 26 October 2007 (CDT)

Representation via matrices?

Just wondering if anyone had already considered an alternative version of the formal definition by defining complex numbers as being a subset of GL_2(R)? Many "university level" people will have seen the basic definition of matrix multiplication, and as such, it may seem less foreign than the definition of multiplication for ordered pairs of real numbers. ...said Barry R. Smith (talk) (Please sign your talk page posts by simply adding four tildes, ~~~~.)

Move some topics to advanced page?

Near the top of the discussion, where a plan for the "complex number" page was sketched, the following comment was made: "I like to introduce complex numbers to my students with the example of the resolution of the cubic equation $x^{3}=15x+4$ with the so called Gerolamo Cardano's method (in fact it is due to Scipione del Ferro and Niccolò Tartaglia). Computations are quite easy, and the striking fact is that during them, one has to use some imaginary number which square would be -1, but once the computations are finished, one gets the three real solutions of the equation!" While it is true that this is probably the earliest example where it became clear that complex numbers were necessary even for the study of real quantities, I definitely disagree with the statement "computations are quite easy". I am also surprised that you introduce complex numbers to students with this example. I have given a project to students of working through this example, and I would not say that they found it easy. In another section of this discussion is the comment, "I certainly agree that articles, especially articles about basic topics like complex numbers, shouldn't scare the reader away right off the bat, but perhaps we need to temper our desire to make the article start out slowly and in a non-intimidating fashion with a bit of logical coherence." It seems to me that our first example might scare many readers away right off the bat. What does a mathematician think about this example? Is it a struggle to get through? Does it make you not want to continue?

The initial idea of writing x=u+v where u and v will be specified later is offputting to many people. This is followed by an application of the binomial theorem, and then an unmotivated factoring step. Then, it is stated that "we only required that x = u + v. Hence, we can choose another condition on u and v. We pick this condition to be 3uv − 15 = 0". Again, the average non-specialist, I would imagine, would wonder, why this condition? And why are we allowed to choose another condition? I could go on listing more potential difficulties that I see in this example.

As we now have the advanced subpage option, why not move this example to an advanced subpage, and then refer to it in the actual article. Perhaps say something like, "A common question is why bother with complex numbers when real numbers almost always seem sufficient for applications. Indeed, the ancients would ignore complex solutions to quadratic equations. It wasn't until the 16th century that it began to be clear that sometimes, complex numbers were indispensable even in problems that seemingly only involve real numbers. An example of this can be found on the "advanced subpage".

Even if this approach isn't taken, might I suggest an alternative way to formulate the current example that is probably more palatable to the average reader: do not derive the solution of the "reduced cubic" by introducing u,v, etc. Instead, just give them the formula for the roots of a reduced cubic -- it isn't very complicated, and an analogy can be made with the quadratic formula. Then show that 4 is a solution, but that the formula gives an expression involving a complex number. Finally, show that this complex can be written as 4.

Also, I think that no matter what, some reference should be made to the fact that although complex numbers are typically introduced these days in high school, when the quadratic formula comes up, that the ancients had versions of the quadratic formula but still didn't accept complex numbers. Barry R. Smith 13:34, 30 April 2008 (CDT)

All sounds good. Go for it! J. Noel Chiappa 22:26, 4 May 2008 (CDT)

New philosophical addition

With regards to Christopher Reiss's new philosophical addition, I believe that this material definitely should be moved out of the introductory paragraph -- as far as I know, an introductory paragraph should usually serve as an abstract for the article, giving a concise non-technical summary of what most would consider the most salient features of the topic?

Now the issues raised are partially addressed already in the history section, notably, the fact that ancients did not believe that complex numbers were "real". Furthermore, the long example in the "advanced" subpage describes the first instance in which complex numbers seemed to be necessary for something. If this material could be improved using some of Christopher's ideas, then I propose that the material be integrated into a whole within to history section.

The only content I would take issue with is the statement that "It is now understood that arithmetic is a pure abstraction which we are free to modify", and related declarations. This is probably the most common philosophical view taken by current mathematicians, and probably many people well versed in mathematics. However, this idea is much older -- Platonic realism is an early example, due to Plato millennia ago. So the phrase "it is now" is misleading. On the other hand, the statement "understood that arithmetic is a pure abstraction" is also misleading. Better would be to say that the majority of current mathematicians believe this. But there is still a raging debate about this in the philosophy of mathematics. For alternatives, see "empiricism" and "fictionalism", for example, at the philosophy of math wikipedia site. Even if we add this material to what Christopher wrote, I believe that content would more properly be put in a philosophy of math site, as the result would probably be quite lengthy and too much of a tangent from the topic of "complex numbers".

Any thoughts?Barry R. Smith 12:20, 22 May 2008 (CDT)